This paper introduces a method to synthesize a 3D tensor field within a constrained geometric domain represented as a tetrahedral mesh. Whereas previous techniques optimize for isotropic fields, we focus on anisotropic tensor fields that are smooth and aligned with the domain boundary or user guidance. The key ingredient of our method is a novel computational design framework, built on top of the symmetric orthogonally decomposable (odeco) tensor representation, to optimize the stretching ratios and orientations for each tensor in the domain. In contrast to past techniques designed only for isotropic tensors, we demonstrate the efficacy of our approach in generating smooth volumetric tensor fields with high anisotropy and shape conformity, especially for the domain with complex shapes. We apply these anisotropic tensor fields to various applications, such as anisotropic meshing, structural mechanics, and fabrication.

Vector image representation methods that can faithfully reconstruct objects and color variations in a raster image are desired in many practical applications. This article presents triangular configuration B-spline (referred to as TCB-spline)-based vector graphics for raster image vectorization. Based on this new representation, an automatic raster image vectorization paradigm is proposed. The proposed framework first detects sharp curvilinear features in the image and constructs knot meshes based on the detected feature lines. It iteratively optimizes color and position of control points and updates the knot meshes. By using collinear knots at feature lines, both smooth and discontinuous color variations can be efficiently modeled by the same set of quadratic TCB-splines. A variational knot mesh generation method is designed to adaptively introduce knots and update their connectivity to satisfy the local reconstruction quality. Experiments and comparisons show that our framework outperforms the existing state-of-the-art methods in providing more faithful reconstruction results. In particular, our method is able to model undetected features and subtle or complicated color variations in-between features, which the previous methods cannot handle efficiently. Our vectorization representation also facilitates a variety of editing operations performed directly over vector images.

The fundamental motivation of the proposed work is to present a new visualization-guided computing paradigm to combine direct 3D volume processing and volume rendered clues for effective 3D exploration. For example, extracting and visualizing microstructures in-vivo have been a long-standing challenging problem. However, due to the high sparseness and noisiness in cerebrovasculature data as well as highly complex geometry and topology variations of micro vessels, it is still extremely challenging to extract the complete 3D vessel structure and visualize it in 3D with high fidelity. In this paper, we present an end-to-end deep learning method, VC-Net, for robust extraction of 3D microvascular structure through embedding the image composition, generated by maximum intensity projection (MIP), into the 3D volumetric image learning process to enhance the overall performance. The core novelty is to automatically leverage the volume visualization technique (e.g., MIP - a volume rendering scheme for 3D volume images) to enhance the 3D data exploration at the deep learning level. The MIP embedding features can enhance the local vessel signal (through canceling out the noise) and adapt to the geometric variability and scalability of vessels, which is of great importance in microvascular tracking. A multi-stream convolutional neural network (CNN) framework is proposed to effectively learn the 3D volume and 2D MIP feature vectors, respectively, and then explore their inter-dependencies in a joint volume-composition embedding space by unprojecting the 2D feature vectors into the 3D volume embedding space. It is noted that the proposed framework can better capture the small/micro vessels and improve the vessel connectivity. To our knowledge, this is the first time that a deep learning framework is proposed to construct a joint convolutional embedding space, where the computed vessel probabilities from volume rendering based 2D projection and 3D volume can be explored and integrated synergistically. Experimental results are evaluated and compared with the traditional 3D vessel segmentation methods and the state-of-the-art in deep learning, by using extensive public and real patient (micro- )cerebrovascular image datasets. The application of this accurate segmentation and visualization of sparse and complicated 3D microvascular structure facilitated by our method demonstrates the potential in a powerful MR arteriogram and venogram diagnosis of vascular disease.